document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085902>Question 1085902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear Tutor,<BR>\n" );
document.write( "please help me.\r<BR>\n" );
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document.write( "20 students take a test with three questions A,B and C. 16 students answer at least one question correctly. 10 answer question A correctly, 8 answer question B correctly, 6 answer question C correctly. 3 students answer both A and B correctly, 4 answer A and C correctly and 1 student answers all three questions correctly. How many students answer questions B and C correctly?\r<BR>\n" );
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document.write( "Could you please explain steps taken while solving this problem?\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700003 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700003\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the basic formula says:\r<BR>\n" );
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document.write( "A or B or C = A + B + C - AB - AC - BC + ABC\r<BR>\n" );
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document.write( "in your problem, this becomes:\r<BR>\n" );
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document.write( "A or B or C = 10 + 8 + 6 - 3 - 4 - 0 + 1 = 18.\r<BR>\n" );
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document.write( "the problem states 20 students, 16 of which answered at least 1 of the problem correctly.\r<BR>\n" );
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document.write( "these numbers indicate that 18 students answered 1 of the problems correctly.\r<BR>\n" );
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document.write( "it does appear that you are missing a category, namely BC.\r<BR>\n" );
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document.write( "i put a 2 in the BC category and then the numbers came out correctly.\r<BR>\n" );
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document.write( "A or B or C = A + B + C - AB - AC - BC + ABC\r<BR>\n" );
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document.write( "with 2 in BC category, formula becomes:\r<BR>\n" );
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document.write( "A or B or C = 10 + 8 + 6 - 3 - 4 - 2 + 1 = 16\r<BR>\n" );
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document.write( "add the 4 that didn't answer any correctly and you get a total of 20.\r<BR>\n" );
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document.write( "bottom line is your problem doesn't come out correctly because there is something wrong with the numbers used.\r<BR>\n" );
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document.write( "i used the following venn diagram generator to confirm that the number don't come out correct as shown, but do come out correct when i put a 2 in the BC category.\r<BR>\n" );
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document.write( "first 2 displays are with numbers as given.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2017/062501.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2017/062502.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "next 2 displays are with numbers as given plus BC category has 2 in it.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2017/062503.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2017/062504.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "to show you how it works logically, i'll use the numbers given, but will add a category of BC = 2.\r<BR>\n" );
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document.write( "you are given, plus what i added:\r<BR>\n" );
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document.write( "total students = 20<BR>\n" );
document.write( "total students who got at least 1 of the problems correct = 16<BR>\n" );
document.write( "total who got none correct = 4 (i created this category).\r<BR>\n" );
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document.write( "A = 10<BR>\n" );
document.write( "B = 8<BR>\n" );
document.write( "C = 6<BR>\n" );
document.write( "AB = 3<BR>\n" );
document.write( "AC = 4<BR>\n" );
document.write( "BC = 2 (this is what i added to make the problem come out corect).<BR>\n" );
document.write( "ABC = 1\r<BR>\n" );
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document.write( "ABC is pure since there are no other categories included in it.\r<BR>\n" );
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document.write( "from AB and AC and BC, subtract ABC pure to get:\r<BR>\n" );
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document.write( "AB pure = 2<BR>\n" );
document.write( "AC pure = 3<BR>\n" );
document.write( "BC pure = 1<BR>\n" );
document.write( "ABC pure = 1\r<BR>\n" );
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document.write( "From A, subtract AB pure, AC pure, ABC pure, to get A pure = 10-2-3-1 = 4\r<BR>\n" );
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document.write( "from B, subtract AB pure, BC pure, ABC pure, to get B pure = 8-2-1-1 = 4\r<BR>\n" );
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document.write( "from C, subtract AC pure, BC pure, ABC pure, to get C pure = 6-3-1-1 = 1\r<BR>\n" );
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document.write( "your pure categories are:\r<BR>\n" );
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document.write( "none pure = 4<BR>\n" );
document.write( "A pure = 4<BR>\n" );
document.write( "B pure = 4<BR>\n" );
document.write( "C pure = 1<BR>\n" );
document.write( "AB pure = 2<BR>\n" );
document.write( "AC pure = 3<BR>\n" );
document.write( "BC pure = 1<BR>\n" );
document.write( "ABC pure = 1\r<BR>\n" );
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document.write( "add them up and you get a total of 20, as you should.\r<BR>\n" );
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document.write( "you can do it by formula or you can do it by logic.\r<BR>\n" );
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document.write( "the total should be 20 students.\r<BR>\n" );
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document.write( "it is, but only after i added the category of BC with 2 in it.\r<BR>\n" );
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document.write( "bottom line:\r<BR>\n" );
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document.write( "you won't get a good answer with the numbers as given, but you will get a good answer with the number as given plus the numbers i added into the BC category.\r<BR>\n" );
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document.write( "some reference that you might find helpful.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.easycalculation.com/algebra/venn-diagram-3sets.php\" target = \"_blank\">https://www.easycalculation.com/algebra/venn-diagram-3sets.php</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.thoughtco.com/probability-union-of-three-sets-more-3126263\" target = \"_blank\">https://www.thoughtco.com/probability-union-of-three-sets-more-3126263</a>\r<BR>\n" );
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